Two-Dimensional Vector Linear Network Coding
In this work, we introduce a new framework of linear network coding (LNC) schemes called two-dimensional (2D) vector LNC, which subsumes conventional (1D) vector LNC as a special case. The new framework models every data unit as an L_{1} \times L_{2} matrix, enabling a new encoding dimension so that...
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| Vydáno v: | Proceedings / IEEE International Symposium on Information Theory s. 1 - 6 |
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22.06.2025
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| ISSN: | 2157-8117 |
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| Abstract | In this work, we introduce a new framework of linear network coding (LNC) schemes called two-dimensional (2D) vector LNC, which subsumes conventional (1D) vector LNC as a special case. The new framework models every data unit as an L_{1} \times L_{2} matrix, enabling a new encoding dimension so that data units can be coded along both row and column dimensions. Compared with (1D) vector LNC of block length L_{1} L_{2}, L_{1} \times L_{2} 2D vector LNC not only reduces the size of local encoding kernels from L_{1} L_{2} \times L_{1} L_{2} to L_{1} \times L_{1} and L_{2} \times L_{2} , but also exhibits greater scalability in the selection of local encoding kernels. In addition, we prove an equivalence between the linear solvability of L_{1} \times L_{2} 2D vector LNC and (1D) vector LNC of block length L_{1} L_{2} . Under the framework of 2D vector LNC, we further formulate 2D circular-shift-based LNC. For distinct odd primes L_{1} and L_{2} , we prove that a network has a (1D) circular-shift-based linear solution of block length \left(L_{1}-1\right)\left(L_{2}-1\right) if and only if it has an \left(L_{1}-1\right) \times\left(L_{2}-1\right) 2D circular-shift-based linear solution. We also demonstrate through an explicit example that, compared with a (1D) circular-shift-based linear solution of block length 8, a 2 \times 4 2D circular-shift-based linear solution requires 23 % fewer XORs in the coding process at intermediate nodes and 13 % fewer XORs in the decoding process. |
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| AbstractList | In this work, we introduce a new framework of linear network coding (LNC) schemes called two-dimensional (2D) vector LNC, which subsumes conventional (1D) vector LNC as a special case. The new framework models every data unit as an L_{1} \times L_{2} matrix, enabling a new encoding dimension so that data units can be coded along both row and column dimensions. Compared with (1D) vector LNC of block length L_{1} L_{2}, L_{1} \times L_{2} 2D vector LNC not only reduces the size of local encoding kernels from L_{1} L_{2} \times L_{1} L_{2} to L_{1} \times L_{1} and L_{2} \times L_{2} , but also exhibits greater scalability in the selection of local encoding kernels. In addition, we prove an equivalence between the linear solvability of L_{1} \times L_{2} 2D vector LNC and (1D) vector LNC of block length L_{1} L_{2} . Under the framework of 2D vector LNC, we further formulate 2D circular-shift-based LNC. For distinct odd primes L_{1} and L_{2} , we prove that a network has a (1D) circular-shift-based linear solution of block length \left(L_{1}-1\right)\left(L_{2}-1\right) if and only if it has an \left(L_{1}-1\right) \times\left(L_{2}-1\right) 2D circular-shift-based linear solution. We also demonstrate through an explicit example that, compared with a (1D) circular-shift-based linear solution of block length 8, a 2 \times 4 2D circular-shift-based linear solution requires 23 % fewer XORs in the coding process at intermediate nodes and 13 % fewer XORs in the decoding process. |
| Author | Ma, Lan Sun, Qifu Tyler Li, Zongpeng |
| Author_xml | – sequence: 1 givenname: Lan surname: Ma fullname: Ma, Lan organization: School of Computer & Communication Engineering, University of Science and Technology Beijing,P. R. China – sequence: 2 givenname: Qifu Tyler surname: Sun fullname: Sun, Qifu Tyler email: qfsun@ustb.edu.cn organization: School of Computer & Communication Engineering, University of Science and Technology Beijing,P. R. China – sequence: 3 givenname: Zongpeng surname: Li fullname: Li, Zongpeng organization: Institute for Network Sciences and Cyberspace, Tsinghua University,P. R. China |
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| Snippet | In this work, we introduce a new framework of linear network coding (LNC) schemes called two-dimensional (2D) vector LNC, which subsumes conventional (1D)... |
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| SubjectTerms | 2D vector linear code circular-shift linear code Data models Decoding Encoding Kernel Linear codes linear solution Network coding Scalability Two-dimensional displays vector linear code Vectors |
| Title | Two-Dimensional Vector Linear Network Coding |
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